Acta Mathematica Academiae Scientiarum Hungaricae 63. (1994)

1994 / 1. szám - Joó I.: Arithmetic functions satisfying a congruence property

I. JOO Let P, N denote the set of all primes resp. positive integers. For any subsets X,Y of N we shall denote by K(P,X,Y) the set of all integer­valued multiplicative functions /(n) for which (1.4) P(E)f(n-\-m) = P(E)f(m) (mod n) holds for every n 6 X and m £ Y. It is obvious that (1.5) fa(n):=na (n = 1,2,...) is a solution of (1.4) for every non-negative integer a and for every triplet (P,X,Y). In this case P(x) = 1 for example, from the result of Subbarao, we have tf(P,N,N) = {/o,/i,/2,...}, where fa is defined in (1.5). Recently, some authors were interested in characterizing all those triplets (P,X,Y) for which (1-6) K(P,X,Y) = {/о,/ь/2) • • •}• In [11]—[14] В. M. Phong obtained some results concerning this problem. He proved that (1.6) holds for the following cases: (i) P(x) = 1, X = N, Y = P; (ii) P(x) = (x - l)fc, x + N, Y = P; (iii) P(x) = xM - 1, X = N, Y = P, where k,M are fixed positive integers. В. M. Phong and J. Fehér in [16] improved the results of Subbarao and Iványi mentioned above showing that (1.6) also holds for P(x) = 1, X = N, Y = {B} with some positive integer B. In [4] we asked for a characterization of those integer-valued multiplica­tive functions /(n) which satisfy (1.7) /(An + B) = C (mod n) for every n £ N, where A ^ 1, b ^ 1 and C/O are fixed integers. We considered this problem with A £ P, proving that there are a non-negative integer a and a real-valued Dirichlet character \A (mod A) such that /(n) = Ха(п)п° for all n € N which are prime to A. The general case has been proved by B. M. Phong [15]. For a fixed integer к ^ 1 we define Nk{n) := m if n = mkh and h is a k­­free power number. In [5] we proved that if an integer-valued multiplicative function /(n) satisfies the congruence f(n + B) = f(B) (mod Nk(n)) Acta Mathematica Hungarica 63, 1994

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